The role played by optical instruments in the advent of the contemporary age is seldom
taken into account. However the telescope has been decisive in the affirmation of the
Copernican universe claimed by Galileo. Without the evidence yielded by this instrument
during the observation of celestial bodies, the clash between the geocentric and
heliocentric conceptions might have continued endlessly. Also the microscope performed a
similar revolutionary function in biology and medicine, opening immense horizons for them.
Before the appearance of the camera, the world descriptions were produced by artists only.
They were valuable representations in which, however, the artist's subjectivity modified
the reality. The camera introduced a much harsher way of observing the world, but much
more objective.

Already from these short considerations, it is possible to guess how great the role
played by optical instruments was in the formation of the world we know. These instruments
are perfectly suitable to the modern and objective way of observing reality, but how do
they work? Today we are in continuous contact with optical instruments and with their
products such as pictures. Understanding the properties of lenses is fundamental to
becoming familiar with these instruments, to use them with confidence and to use lenses in
a creative manner in order to design optical instruments. This is exactly what we shall do
in following articles and for that it is necessary to have a basic knowledge of optics.
Starting from the complicated theoretical descriptions in a physics book, the
understanding of lens properties is not easy. However, by means of some simple experiments
instead, it is possible to overcome many abstract obstacles. At this point, the return to
the physics text will be simpler and more effective.

There are two types of lenses: the convergent ones and the divergent ones.
The convergent lenses are able to converge the light of the Sun until to form a
little and very bright disk that is the image of our star, while the divergent
lenses diverge the bundle of light coming from the Sun so its image cannot form. Here we shall deal
only with convergent ones, which are more important. The first experiments we shall
perform now are intended to show the main properties of convergent lenses. The last one,
through the combined use of two lenses, will show how some important optical instruments
such as the telescope and compound microscope work.
A converging lens can be used in two main ways: as an image producer and as a magnifier.

Equipment: a convergent lens with focal length between 100
and 300 mm, a candle, a white box, a meter rule. Buy the lens in an optical or
photographic shop. Clear a table and prepare an "optical bench" like the one
showed in figure 1. The p and q distances must be greater than the lens focal length.
Light the candle and switch off the light. Modify the p and q values, until the candle
image appears distinct on the box which you are using as a screen. Perform multiple tests,
changing the distances. Try also exchanging the p and q distances.

How is the image formed? In order to explain this, normally two fundamental properties
of lenses are taken into account:
- deviating a light beam parallel to its own axis, then making it to pass through the
focus;
- leaving unaltered the path of the rays which pass through the lens center.

With reference to figure 2, take into account any object
point, for convenience the extreme one. Among all light rays starting from this point,
there are 3 whose path is particularly easy to follow. The A ray passing through the lens
center and which is not deflected; the B ray which comes to the lens moving parallel to
the axis and which passes through F1; the C ray which in a similar way passes through F2
and leaves the lens parallel to the optical axis. These three rays form an image point
where they cross one another.

Operating in the same way for the other object points, you obtain the whole image. To
trace these schemes, only two of these rays are required. There are also other rays, not
parallel to the axis and not passing through the focuses, which contribute to the image
formation. Also for these it would be possible to calculate the ray path, but to describe
how a lens works, the ones we have taken into account are sufficient.

During this experiment, you will see that the image formed is inverted. This can be
easily explained following the A ray path. In fact a ray starting from a high position on
the object, after passing through the lens center, will be inverted on the image side.

While performing experiments like those described in the
previous paragraph, measure the height of the object and that of its image (fig. 3). Since
the candle flame does not have a stable image, replace the candle with an object well lit
by a lamp as shown in figure 5. If it is necessary, mask any stray light which does not
pass through the lens, so as to obtain a higher contrast and a more visible image.

The size of the image is not invariable, in fact as the lens is moved towards the
object, the image moves out and becomes larger (therefore you must move the screen away).
The magnification is given by I=H/h, where H is the height of the image and h the one of
the object.

It is not always possible to measure those dimensions. For example we cannot open a
camera with the film inside, to measure the image. It is difficult even to measure very
distant or too small an object. In these cases, the magnification can be determined by
measuring the distances p and q. In fact, for thin lenses, the ray passing through the
lens center and which is not deflected (fig. 3), contribute to forming two similar
triangles which have a common vertex at the lens center. On the basis of the properties of
similar triangles H/h=q/p, and, since I=H/h, also I=q/p. Verify this relation
experimentally.

As you bring the the lens towards the illuminated object, you come to a position in
which the image is far away. If the lens object distance is equal to the focal length, the
image will be formed at infinity, whereas an object placed at infinity will produce its
own image at the focal point. Furthermore, a lens placed at 2F from the object, will form
the image at the same 2F distance. In this case, the magnification ratio is equal to 1.

What is the focal length? This word comes from the Latin "focus" (fire) for
the lens' property of concentrating the sunlight so much as to set fire to combustible
objects. The distance from the lens at which those objects must be kept has been named
focal length. In optics this word is defined as the distance from the lens node (we will
see that later) to the point at which a ray, which was initially parallel to the optical
axis, intercepts the axis after being deflected by the lens.
To determine the focal length of a converging thin lens, use again your special optical
bench. Arrange the illuminated object and the lens in such a way as to obtain a sharp
image on the screen. Measure the p and q distances with the meter rule. The focal length
is given by:

1 1 1 p x q
--- = --- + --- in explicit form: F = -------
F p q p + q

To obtain a better approximation, more measurements must be made to calculate the
average value of the lens focal length.

There is another way to indicate the focal length of a lens. In the fields of the
production and the market of eyeglasses, instead of focal length people prefer speak of
lens power, measured in diopters. So, if you have to buy an eyeglass lens, you need to
know its power. Focal length and power of a lens are bound to each other and you can
easily pass from one to the other using this simple formula:

D = 1/FL

where:
D = diopters
FL = lens focal length (expressed in meters!)
Besides, people place the sign "+" before the power of a converging lens and the
sign "-" before the power of a diverging lens.

Let's make a couple of examples:
- a converging lens of half a meter of focal length has a power of +2 diopters. In fact: D
= 1/0,5 = +2
- a diverging lens of 4 meter of focal length has a power of -0.25 diopters. In fact: D =
1/-4 = -0,25

Equipment: a convergent lens with focal length included between 20 and 60
mm.
1) Observe with the naked eye an object placed at a distance of 250 mm;
2) observe the same object with the lens and compare the two images.
The lens must be kept close to the eye. If it is planoconvex, keep the plane surface
towards the eye. Approach the object until it becomes distinct.

This experiment is very simple. But how does the lens magnify the object?
The nearest distance of distinct vision with the naked eye is considered to be 250 mm. A
normal adult man has difficulty seeing clearly objects closer than 250 mm. Converging
lenses allows us to approach the object well below this distance and to still see it
clearly. As we approach the object we will see it larger (fig. 4). A human eye is able to
work with parallel light (from distant objects) or with light of limited divergence
(objects not nearer than 250 mm). Converging lenses reduce the divergence of rays coming
us from an object nearer than 250 mm, and allows us to still see it clearly.

The object to be observed must be placed between the front focus (F2) and the lens
(fig. 4). For convenience we assume that the optical center of the eye coincides with the
back focus (F1) of the lens. (The distance of the eye from the lens is not important, but
in practice we will keep the eye close to the lens). Let's consider an object point.
Among all the rays leaving the object we shall take for convenience ray A parallel to the
axis, which is deflected by the lens and passes through the back focus F1 and arrives at
the retina. We shall also take ray B passing through the lens center which is not
deflected, and enters the eye where it is deflected by the cornea and intercepts ray A on
the retina, forming an image point. The image formed on the retina is seen in a plane
conventionally placed at a distance of 250 mm from the eye. It is not a real image, in the
sense that it cannot be recorded on film and for this reason it is called virtual.
This image is perceived the right way up, although in the eye it is upside-down. Even when
we are not using lenses, the images formed in the eye are inverted. It is the brain that
corrects this image.

At the onset the A and B rays have a great divergence; on the other side of the lens,
their divergence is reduced. If the object were placed in F2, the lens would make the A
and B rays parallel, and to see the image clearly, the eye would focus at infinity.
Finally, as we were saying, the magnifying lens reduces the divergence of the light coming
from a close object. The lens also allows the object to be viewed clearly and magnified
even below 250 mm.
Notice that the same converging lens can be used both as a magnifier and as an image
generator. Note that the lens producing images turns them upside-down, while the
magnifying glass keeps them the correct way up.

In the case of the magnifying lens, the magnification power is determined by the
following relation: I=250/F, where F is the lens focal length (mm) and 250 is the
conventional distance for distinct vision or reading. For example, a lens with 50 mm focal
length will magnify 5 times. This is valuable when the eye is focused at infinity,
whereas when it is focused for near vision, the relation becomes I=(250/F)+1.
Hence, the lens of 50 mm focal length magnifies from 5 to 6 times, according to the eye's
accommodation.
In a previous article, in which we talked about a little glass-sphere microscope, you
could see to what extent a lens can magnify. However it is necessary to say that this is
an extreme situation: normally a magnifying glass does not exceed 20 X.

Now you are finally ready for the conclusive experiment, the one that should enable you
to understand how some of the most important optical instruments work. Let's go back to
the optical bench. However, this time replace the box with a translucent screen. You can
make it with a card frame on which you have fixed a piece of white plastic taken from a
plastic bag (fig. 5). Focus the image on the screen and you can observe the image
appearing from behind the screen.

You can also enlarge the image with a magnifying glass by
taking the lens you used for the previous experiment and observing the image behind the
translucent screen. As you can see, the image appears magnified. So far there is nothing
strange. While you continue to watch the upside- down image, try to move the screen a
little. The image keeps steady. Oh, dear! Then...
Remove the screen. Miracle! The image stays there. It is "floating" in the
space. Therefore the screen was useless! It actually was! Not only is the image clearer
and brighter, it is colored and in 3D too.

There, you have built a
telescope! The lens nearest the object is your objective, the one near the eye is the
eyepiece (fig.6).

Continuing this experiment, if the objective is brought closer to the
object, the image moves away and becomes larger. Regulate the p and q distances in such a
way that the image becomes larger than the object. Observe it with the magnifying lens: in
this way you have obtained a compound microscope (fig. 7).

So, what distinguishes a microscope from a telescope? As you can see, the
optical structure is the same, but with telescopes objects are distant whereas in a
microscope they are close. Normally a telescope observes objects typically placed at
hundreds of meters or more, and a microscope observes objects placed at a few millimeters
or less from the objective.

Up to now, we have dealt with thin lenses. This last paragraph introduces the concept
of nodes and gives you an idea of the type of errors which occur when thin lens formulas
are applied to real lenses. Thin lenses are considered to have no thickness, hence it is
considered that the ray paths deviate when they meet the plane of the lens (fig. 8/A). In
reality, the ray path is rectilinear inside an homogenous medium, and deflects when
entering another medium with a different refractive index. Therefore, a ray passing
through a lens is bent when it enters the glass and bent again when it leaves the lens
(fig. 8/B).

In order to introduce the concept of a node, let's consider
the ray which, among all those entering a real lens, does not deviate as it passes through
the lens (fig. 8/C), and links the entering and leaving point of incident and emerging
rays. Extending the path of both external rays to their intersection with the optical
axis, locates two point which are named nodes. The focal length of a lens is
referred to as the closer node.

The nodes of real lenses and their distance are neglected by thin lens
formulas and this leads to some errors. However, for first drawing an optical instrument,
or whenever one does not need high accuracy, this error is small and can be accepted in
exchange for simpler calculations. In fact, the formulas for thin lenses are widely used
by opticians for preliminary calculations and analysis, and are of value in most cases.
But, especially if you are dealing with thick lenses of short focal length, or if you need
precision, you must refer to formulas for thick lenses which you can find in optics texts.

The converging lens can:
- produce a real image of an object;
- magnify the apparent dimensions of an object or an image;
- be used with other lenses to build optical instruments.

I hope that these simple experiments have been able to introduce you to the world of
optics. Now what do you need to make a telescope or a microscope? Just a bit of spirit of
adventure. If the first instruments will be made up with "bad" lenses, no
matter, on the contrary it is an important step to understand why in these instruments the
objectives and the eyepieces are formed by more than one lens, but this is another story.